Why can't I do a loop cut after Merging these vertices?
The fundamental problem is that loop cuts work on quads; they are interrupted by triangles, and faces with more than four edges. You can do the loop cut before you merge all the points together, because Blender treats each face as a quad, that is a face defined by four vertices, where one edge has zero length because two of the vertices have the same co-ordinages. The merge makes all of quads triangles, because two of the vertices on each face are combined into one. Similarly, if all of the three or more vertices of a face have the same co-ordinates, you still have a face, even though the face has zero area.

With respect to achieving an effect similar to the loop cuts, switch to edge select mode, select all the radial edges (but not the circumferential ones), and use the subdivide tool, setting the number of subdivisions to the desired value. Another way is to use the knife tool. Yet another way is to extrude the vertices around the circumference, and scale them almost to the center, then extrude again, and scale the second set of vertices to the center point, and merge the center vertices together. Now you can still make your loop cuts in the quad faces created by the first extrusion, and you can select the edge loop created by the second extrusion and use the edge slide to position it where you want it.